Currently, private banks and others are in such dire straights that they are willing to swallow the stigma and borrow en masse from the Federal Reserve. The scale and rate of increase of borrowing from the Fed suggest perhaps some of the financial institutions currently at the trough have no other way to survive.
Despite the alarmist introduction, there are several reasons that the sight of such a plot has not yet caused me to cash out my 401(k) and put it all into shotgun shells and iodine pills.
1. The nature of Federal Reserve Funding has changed since January of 1919. In particular, my Grandparents only had the discount window. Currently, financial institutions have access to some or all of (depending on their status with the Fed): the TAF, TSLF, PDCF, CPFF, MMIFF and TALF. All of these lending facilities were designed to expand the ability of the Federal reserve to lend money in ways that it had previously never done. Of course, the fact that these programs were implemented suggests that at least someone was of the opinion that the economy is in a place that it has never been before.
2. Stigma is fickle. If everyone is borrowing and the rates are good, then even sound banks should choose to borrow.
3. At least we know where some of the money is going. The graph below gives excess reserves for Federal Reserve depository institutions.
Relative to the current market, Fed reserves offer an attractive risk/return profile. Additionally, there are probably many banks out there with significant CDS exposure that are hoarding reserves in the event that there are future credit events or the need to post additional collateral on their own positions (due to a worsening of their own rating).
Either way, all of those reserves could buy a lot of shotgun shells and iodine pills.

Stigma is fickle. We have the discount window for a reason.